Asymptotic form conjecture for partitions of a 3-row rectangle

Let $p(3,n)$ denote the number of partitions of a $3\times n$ rectangle into permitted rectangular blocks, as defined in the paper. Asymptotic form conjecture. There exist constants $k_3,\lambda_3>0$ and $\delta_3\in\mathbb{R}$ such that

$$p(3,n)\sim k_3n^{\delta_3}\exp\bigl(\lambda_3\sqrt{n}\bigr)\text{ as }n\to\infty.$$

The conjecture is motivated by the established asymptotic for $p(2,n)$ and predicts an analogous exponential-polynomial asymptotic for partitions of a $3\times n$ rectangle. No resolution is supplied in the source.